Effects of complex parameters on classical trajectories of Hamiltonian systems

Anderson $\textit{et al}$ have shown that for complex energies, the classical trajectories of $\textit{real}$ quartic potentials are closed and periodic only on a discrete set of eigencurves. Moreover, recently it was revealed that, when time is complex $t$ $(t=t_{r}e^{i\theta _{\tau }}),$ certain real hermitian systems possess close periodic trajectories only for a discrete set of values of $\theta _{\tau }$. On the other hand it is generally true that even for real energies, classical trajectories of non $\mathcal{PT}$- symmetric Hamiltonians with complex parameters are mostly non-periodic and open. In this paper we show that for given real energy, the classical trajectories of $\textit{complex}$ quartic Hamiltonians $H=p^{2}+ax^{4}+bx^{k}$, (where $a$ is real, $b$ is complex and $k=1$ $or$ $2$) are closed and periodic only for a discrete set of parameter curves in the complex $b$-plane. It was further found that given complex parameter $b$, the classical trajectories are periodic for a discrete set of real energies (i.e. classical energy get discretized or quantized by imposing the condition that trajectories are periodic and closed). Moreover, we show that for real and positive energies (continuous), the classical trajectories of $\textit{complex}$ Hamiltonian $H=p^{2}+\mu x^{4}, (\mu=\mu _{r}e^{i\theta })$ are periodic when $\theta =4 tan^{-1}[(n/(2m+n))]$ for $\forall$ $n$ and $m\in \mathbb{Z}$.Comment: 9 pages, 2 tables, 6 figure

Explicit energy expansion for general odd degree polynomial potentials

In this paper we derive an almost explicit analytic formula for asymptotic eigenenergy expansion of arbitrary odd degree polynomial potentials of the form $V(x)=(ix)^{2N+1}+\beta _{1}x^{2N}+\beta _{2}x^{2N-1}+\cdot \cdot \cdot \cdot \cdot +\beta _{2N}x$ where $\beta _{k}^{\prime }$s are real or complex for $1\leq k\leq 2N$. The formula can be used to find semiclassical analytic expressions for eigenenergies up to any order very efficiently. Each term of the expansion is given explicitly as a multinomial of the parameters $\beta _{1},\beta _{2}....$ and $\beta _{2N}$ of the potential. Unlike in the even degree polynomial case, the highest order term in the potential is pure imaginary and hence the system is non-Hermitian. Therefore all the integrations have been carried out along a contour enclosing two complex turning points which lies within a wedge in the complex plane. With the help of some examples we demonstrate the accuracy of the method for both real and complex eigenspectra.Comment: 10 page

Pre-pausal devoicing and glottalisation in varieties of the south-western Arabian peninsula

A wide range of modern Arabic dialects exhibit devoicing in pre-pausal (utterance-final) position. These include Cairene [20], Gulf Arabic, San’ani [8], [18], Manaxah [19], Central Highland Yemeni dialects [1], Rijal Alma‘ (Asiri p.c.), Central Sudanese (Dickins p.c.), Çukurova [15], Kinderib [9], E. Fayyum [2]. In some dialects, pausal devoicing is reported to be accompanied by aspiration (e.g. Cairene, [19]), in others by glottalisation (e.g. Fayyum, [2]; Manaxah, [18]; San’ani, [8], [18]). As preliminary work to a study of pausal phenomena in the south-western Arabian Peninsula, we examine data from two Arabic dialects – San’ani (SA), spoken in the Old City of San’a, Yemen, and the Asiri dialect of Rijal Alma‘ (RA) – and from Mehriyōt, an eastern dialect of the modern south Arabian language, Mehri, spoken in Yemen. We begin by presenting a summary of pausal phenomena in SA. We then consider the behaviour of final oral stops – velar, coronal and labial – final coronal fricatives, final nasals and liquids, and final vowels. Initial comparison with data from RA and Mehriyōt indicates that utterance-final devoicing is more advanced in SA than in the other varieties, and involves a greater range of segment types. The first set of pausal examples were extracted from Watson’s recordings of spontaneous SA monologues on the Semitic Spracharchiv. The main speaker is a young semi educated woman.1 Those forms which exist as lexemes in RA, plus lexemes involving similar pre-pausal segments in comparable syllable types, were recorded utterance-finally by Yahya Asiri, a native speaker of RA. Pausal forms for Mehriyōt were extracted from the late Alexander Sima’s recordings of spontaneous speech on the Semitic sound archive [16]. The Mehriyōt speaker is a low- to semi-educated early middle-aged man. Data were analysed using the phonetic analysis programme PRAAT (www.praat.org)

3,4-Dimethyl-N-[(E)-3-nitro­benzyl­idene]-1,2-oxazol-5-amine

In the title compound, C12H11N3O3, the dihedral angle between the 3-nitro­benzaldehyde and 5-amino-3,4-dimethyl-1,2-oxazole moieties is 2.46 (12)°. The mol­ecule is close to planar, the r.m.s. deviation for the non-H atoms being 0.028 Å. The packing only features van der Waals inter­actions between the mol­ecules

(2E)-3-[4-(Dimethyl­amino)­phen­yl]-1-(2,5-dimethyl-3-thien­yl)prop-2-en-1-one

The asymmetric unit of the title compound, C17H19NOS, contains two independent mol­ecules which differ in the dihedral angles between the five- and six-membered rings [12.52 (10) and 4.63 (11)°]. Weak inter­molecular C—H⋯O hydrogen bonds link the two independent mol­ecules into pseudocentrosymmetric dimers. In one mol­ecule, the O atom of the carbonyl group is disordered over two positions in a 0.699 (4):0.301 (4) ratio